On multipartite Hajnal-Szemer\'edi theorems

Let $G$ be a $k$-partite graph with $n$ vertices in parts such that each vertex is adjacent to at least $\delta^*(G)$ vertices in each of the other parts. Magyar and Martin \cite{MaMa} proved that for $k=3$, if $\delta^*(G)\ge 2/3n $ and $n$ is sufficiently large, then $G$ contains a $K_3$-factor (a spanning subgraph consisting of $n$ vertex-disjoint copies of $K_3$) except that $G$ is one particular graph. Martin and Szemer\'edi \cite{MaSz} proved that $G$ contains a $K_4$-factor when $\delta^*(G)\ge 3/4n$ and $n$ is sufficiently large. Both results were proved by the Regularity Lemma. In this paper we give a proof of these two results by the absorbing method. Our absorbing lemma actually works for all $k\ge 3$.